while \(\omega = 2(\alpha _2+3\alpha _4)\) .
Here the stretching factor \(B=(\alpha _2+6\alpha _4)\) . If \(\alpha _4=0\) , the solution Eq.(REF ) reduces to the standard rogue wave solution of the NLSE Eq.(REF ) [1]}. In Fig.(REF ), the solution \(a_{34}^{\prime }\) , presented by the red dotted line, come in contact with the \(y\)  axes at \((\kappa _1,\kappa _2)=(0,\sqrt{2})\)  in the first quadrant and intersects at \(x\)  axes at \((\kappa _1,\kappa _2)=(\sqrt{2},0)\) . Both of these points are identified as green rectangle on the \(x\)  and \(y\)  axes. We may investigate the structure of the breather at these points. To do this, first we put \(a_{34}^{\prime }\)  as \(\kappa _2\)  value in Eq.(REF ) that makes the equation defined by a single MI frequency \(\kappa _1\)  with one gain parameter \(V_{H1}\) . If we assign \(\kappa _1=\sqrt{2}\)  that is the green rectangle on the \(x\)  makes the solution undefined as the breather corresponds to \(\kappa _2\)  becomes zero. Similarly, if we put \(\kappa _1=0\) , again the solution becomes undefined. However, we can recover the solution either taking the limit \(\kappa _1 \rightarrow 0\)  or \(\kappa _1 \rightarrow \sqrt{2}\)  in Eq.(REF ). In both of these limits we get a second-order periodic solution:
\(\nonumber G_2&=&-2 \left\lbrace -3+2 \sqrt{2} \cos \left(\sqrt{2} t_{\text{s2}}\right)+4 t_{\text{s1}}^2+x_{\text{s1}}^2\right\rbrace \\H_2&=&4 \left\lbrace -2+\sqrt{2} \cos \left(\sqrt{2} t_{\text{s2}}\right)\right\rbrace  x_{\text{s1}}\\\nonumber D_2&=&\sqrt{2} \cos \left(\sqrt{2} t_{\text{s2}}\right) \left(-7+4 t_{\text{s1}}^2+x_{\text{s1}}^2\right)\\\nonumber &+&2 \left\lbrace 5+4 t_{\text{s1}} \left[-2 \sin \left(\sqrt{2} t_{\text{s2}}\right)+t_{\text{s1}}\right]+x_{\text{s1}}^2\right\rbrace \) 
